Average Error: 9.8 → 0.1
Time: 14.3s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\frac{2}{\left(x - 1\right) \cdot x}}{x + 1}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\frac{2}{\left(x - 1\right) \cdot x}}{x + 1}
double f(double x) {
        double r138045 = 1.0;
        double r138046 = x;
        double r138047 = r138046 + r138045;
        double r138048 = r138045 / r138047;
        double r138049 = 2.0;
        double r138050 = r138049 / r138046;
        double r138051 = r138048 - r138050;
        double r138052 = r138046 - r138045;
        double r138053 = r138045 / r138052;
        double r138054 = r138051 + r138053;
        return r138054;
}
↓
double f(double x) {
        double r138055 = 2.0;
        double r138056 = x;
        double r138057 = 1.0;
        double r138058 = r138056 - r138057;
        double r138059 = r138058 * r138056;
        double r138060 = r138055 / r138059;
        double r138061 = r138056 + r138057;
        double r138062 = r138060 / r138061;
        return r138062;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.8
Target0.3
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 9.8

    \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]

  2. Simplified9.8

    \[\leadsto \color{blue}{\frac{1}{x - 1} - \left(\frac{2}{x} - \frac{1}{x + 1}\right)}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt24.4

    \[\leadsto \frac{1}{x - 1} - \left(\frac{2}{x} - \frac{1}{\color{blue}{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}}}\right)\]

  5. Applied add-cube-cbrt24.4

    \[\leadsto \frac{1}{x - 1} - \left(\frac{2}{x} - \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}}\right)\]

  6. Applied times-frac25.1

    \[\leadsto \frac{1}{x - 1} - \left(\frac{2}{x} - \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{x + 1}}}\right)\]
  7. Using strategy rm

  8. Applied frac-times24.4

    \[\leadsto \frac{1}{x - 1} - \left(\frac{2}{x} - \color{blue}{\frac{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}}}\right)\]

  9. Applied frac-sub28.6

    \[\leadsto \frac{1}{x - 1} - \color{blue}{\frac{2 \cdot \left(\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}\right) - x \cdot \left(\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}\right)}{x \cdot \left(\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}\right)}}\]

  10. Applied frac-sub25.8

    \[\leadsto \color{blue}{\frac{1 \cdot \left(x \cdot \left(\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}\right)\right) - \left(x - 1\right) \cdot \left(2 \cdot \left(\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}\right) - x \cdot \left(\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}\right)\right)}{\left(x - 1\right) \cdot \left(x \cdot \left(\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}\right)\right)}}\]

  11. Simplified25.5

    \[\leadsto \frac{\color{blue}{x \cdot \left(\left(x + 1\right) \cdot 1\right) - \left(x - 1\right) \cdot \left(\left(x + 1\right) \cdot 2 - x \cdot 1\right)}}{\left(x - 1\right) \cdot \left(x \cdot \left(\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}\right)\right)}\]

  12. Simplified25.5

    \[\leadsto \frac{x \cdot \left(\left(x + 1\right) \cdot 1\right) - \left(x - 1\right) \cdot \left(\left(x + 1\right) \cdot 2 - x \cdot 1\right)}{\color{blue}{\left(\left(x - 1\right) \cdot x\right) \cdot \left(x + 1\right)}}\]

  13. Taylor expanded around 0 0.3

    \[\leadsto \frac{\color{blue}{2}}{\left(\left(x - 1\right) \cdot x\right) \cdot \left(x + 1\right)}\]
  14. Using strategy rm

  15. Applied associate-/r*0.1

    \[\leadsto \color{blue}{\frac{\frac{2}{\left(x - 1\right) \cdot x}}{x + 1}}\]

  16. Final simplification0.1

    \[\leadsto \frac{\frac{2}{\left(x - 1\right) \cdot x}}{x + 1}\]

Reproduce

herbie shell --seed 2019208 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64

  :herbie-target
  (/ 2 (* x (- (* x x) 1)))

  (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1))))